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CVE/wip.R

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R

library(microbenchmark)
elem.pairs <- function(elements) {
# Number of elements to match.
n <- length(elements)
# Create all combinations.
pairs <- rbind(rep(elements, each=n), rep(elements, n))
# Select unique combinations without self interaction.
return(pairs[, pairs[1, ] < pairs[2, ]])
}
rStiefl <- function(p, q) {
return(qr.Q(qr(matrix(rnorm(p * q, 0, 1), p, q))))
}
grad <- function(X, Y, V, h, persistent = TRUE) {
n <- nrow(X)
p <- ncol(X)
# Projection matrix onto `span(V)`
Q <- diag(1, p) - tcrossprod(V, V)
# Vectorized distance matrix `D`.
vecD <- rowSums((X_diff %*% Q)^2)
# Weight matrix `W` (dnorm ... gaussean density function)
W <- matrix(1, n, n) # `exp(0) == 1`
W[lower] <- exp((-0.5 / h) * vecD^2) # Set lower tri. part
W[upper] <- t.default(W)[upper] # Mirror lower tri. to upper
W <- sweep(W, 2, colSums(W), FUN = `/`) # Col-Normalize
# Weighted `Y` momentums
y1 <- Y %*% W # Result is 1D -> transposition irrelevant
y2 <- Y^2 %*% W
# Per example loss `L(V, X_i)`
L <- y2 - y1^2
# Vectorized Weights with forced symmetry
vecS <- (L[i] - (Y[j] - y1[i])^2) * W[lower]
vecS <- vecS + ((L[j] - (Y[i] - y1[j])^2) * W[upper])
# Compute scaling of `X` row differences
vecS <- vecS * vecD
G <- crossprod(X_diff, X_diff * vecS) %*% V
G <- (-2 / (n * h^2)) * G
return(G)
}
grad2 <- function(X, Y, V, h, persistent = TRUE) {
n <- nrow(X)
p <- ncol(X)
# Projection matrix onto `span(V)`
Q <- diag(1, p) - tcrossprod(V, V)
# Vectorized distance matrix `D`.
# vecD <- rowSums((X_diff %*% Q)^2)
vecD <- colSums(tcrossprod(Q, X_diff)^2)
# Create Kernel matrix (aka. apply kernel to distances)
K <- matrix(1, n, n) # `exp(0) == 1`
K[lower] <- exp((-0.5 / h) * vecD^2) # Set lower tri. part
K[upper] <- t(K)[upper] # Mirror lower tri. to upper
# Weighted `Y` momentums
colSumsK <- colSums(K)
y1 <- (K %*% Y) / colSumsK
y2 <- (K %*% Y^2) / colSumsK
# Per example loss `L(V, X_i)`
L <- y2 - y1^2
# Compute scaling vector `vecS` for `X_diff`.
tmp <- kronecker(matrix(y1, n, 1), matrix(Y, 1, n), `-`)^2
tmp <- as.vector(L) - tmp
tmp <- tmp * K / colSumsK
vecS <- (tmp + t(tmp))[lower] * vecD
G <- crossprod(X_diff, X_diff * vecS) %*% V
G <- (-2 / (n * h^2)) * G
return(G)
}
n <- 200
p <- 12
q <- 10
X <- matrix(runif(n * p), n, p)
Y <- runif(n)
V <- rStiefl(p, q)
h <- 0.1
pair.index <- elem.pairs(seq(n))
i <- pair.index[1, ] # `i` indices of `(i, j)` pairs
j <- pair.index[2, ] # `j` indices of `(i, j)` pairs
lower <- ((i - 1) * n) + j
upper <- ((j - 1) * n) + i
X_diff <- X[i, , drop = F] - X[j, , drop = F]
stopifnot(all.equal(
grad(X, Y, V, h),
grad2(X, Y, V, h)
))
microbenchmark(
grad = grad(X, Y, V, h),
grad2 = grad2(X, Y, V, h)
)